EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

Demo).
11- Third Compute Coherence as the ratio of the auto-spectra and
cross-spectra
Coherence is usually defined as:
Eq. 12 - Coherence (f) =
Cross − Spectrum(
f ) XY
( Autospectrum(
f )( X ))( Autospectrum(
f )( Y ))
2
However, this standard mathematical definition of coherence hides some of
the essential statistical nature and structure of coherence. To illustrate the
fundamental statistics of coherence let us return to our simple algebraic
notation:
Eq. 13 -
Coherence (f) =
(
∑
N
( a(
x)
u(
y)
+ b(
x)
v(
y)))
∑
N
( a(
x)
2
2
+ b(
x)
2
+ (
)
∑
N
∑
N
( a(
x)
v(
y)
− b(
x)
u(
y)))
u(
y)
2
+ v(
y)
2
)
2
Where N and the summation sign represents averaging over frequencies in
the raw spectrogram or averaging replications of a given frequency or both.
The numerator and denominator of coherence always refers to smoothed or
averaged values, and, when there are N replications or N frequencies then
each coherence value has 2N degrees of freedom. Note that if spectrum
estimates were used which were not smoothed or averaged over frequencies
nor over replications, then coherence = 1 (Bendat and Piersol, 1980;
Benignus, 1968; Otnes and Enochson, 1972). In order to compute
coherence, averaged cospectrum and quaspectrum smoothed values with
degrees of freedom > 2 and error bias = 1/N is used.
The numerical example of coherence used the average cospectrum
and quadspectrum across replications in Table III. For example from Table
III the coherence at 1.25 Hz is:
2
2
0.073 + 0.031
Eq. 14 - Hand Calculator Coherence (1.25 Hz) = = 0. 026
0.586(0.419)